What are the characteristics of the exponential distribution?

In reliability engineering the exponential distribution has been found to adequately model the failure rate of electronics during their useful life [1]. The probability density function (PDF) for the exponential distribution is shown in Figure 1 and is given below by Equation 1. For any time period of interest, t, the area under the PDF curve from t to infinity is equal to the reliability, R(t). The area under the PDF curve to the left of t is equal to the probability of failure (1-R(t)).

                   f(t)=le^-lt                                                                                              [ 1 ]

The cumulative distribution function (area under PDF curve) is obtained by integrating the PDF as shown in Figure 2 and given by Equation 2.

                                                                                               [ 2 ]

Figure 1 - Probability Density Function (Exponential Distribution)

Figure 2 - Cumulative Distribution Function

 The reliability function (probability that a device will not fail), is given by Equation 3.

                   R(t)=1-F(t)==  e^-lt                         [ 3 ]

Figure 3 - Reliability Function (Exponential Distribution)

The hazard function is given by Equation 4. For the exponential distribution, the hazard function, or instantaneous failure rate, is constant over time, as shown in Figure 4. This is one of the fundamental relationships of reliability analysis and allows for the simple addition of component failure rates to obtain equipment or system level failure rates. It is a measure of the change in survivor rate per unit time.

                                                                                   [ 4 ]

Figure 4 - Hazard Function (Exponential Distribution)

[1] MIL-HDBK-338, Electronic Reliability Design Handbook, 15 Oct 84.